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The Landau-Lifshitz-Gilbert equation yields a mathematical model to describethe evolution of the magnetization of a magnetic material, particularly inresponse to an external applied magnetic field. It allows one to take intoaccount various physical effects, such as the exchange within the magneticmaterial itself. In particular, the Landau-Lifshitz-Gilbert equation encodesrelaxation effects, i.e., it describes the time-delayed alignment of themagnetization field with an external magnetic field. These relaxation effectsare an important aspect in magnetic particle imaging, particularly in thecalibration process. In this article, we address the data-driven modeling ofthe system function in magnetic particle imaging, where theLandau-Lifshitz-Gilbert equation serves as the basic tool to include relaxationeffects in the model. We formulate the respective parameter identificationproblem both in the all-at-once and the reduced setting, present reconstructionalgorithms that yield a regularized solution and discuss numerical experiments.Apart from that, we propose a practical numerical solver to the nonlinearLandau-Lifshitz-Gilbert equation, not via the classical finite element method,but through solving only linear PDEs in an inverse problem framework.

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging